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arXiv · 1402.1591

Recent progress in smoothing estimates for evolution equations

Michael Ruzhansky·Mitsuru Sugimoto
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Abstract

This paper is a survey article of results and arguments from several of authors' papers, and it describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on ideas of comparison principle and canonical transforms. For operators $a(D_x)$ of order $m$ satisfying the dispersiveness condition $\nabla a(ξ)\neq0$, a range of smoothing estimates is established. Especially, time-global smoothing estimates for the operator $a(D_x)$ with lower order terms are the benefit of our new method. These estimates are known to fail for general non-dispersive operators. For the case when the dispersiveness breaks, we suggest a modification of the smoothing estimate. It is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator $a(D_x)$. Moreover, it does continue to hold for a variety of non-dispersive operators $a(D_x)$, where $\nabla a(ξ)$ may become zero on some set. It is interesting that this method allows us to carry out a global microlocal reduction of equations to the translation invariance property of the Lebesgue measure.

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Michael Ruzhansky, Mitsuru Sugimoto. 2014-02-07. Recent progress in smoothing estimates for evolution equations. https://doi.org/10.1007/978-3-319-00125-8_13

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